examples_1219

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) -1
• \(h=\) animated
• \(k=\) -5

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) animated
• \(h=\) -4
• \(k=\) 3

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) -1
• \(h=\) animated
• \(k=\) 4

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) animated
• \(h=\) 5
• \(k=\) 1

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) 1
• \(h=\) 5
• \(k=\) animated

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) -1
• \(h=\) animated
• \(k=\) -5

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) -1
• \(h=\) animated
• \(k=\) -4

Try it out in Desmos!

In Desmos graphing calculator, define \(f(x)\) with the following parameterized equation, and add sliders for all three parameters (\(a\), \(h\), and \(k\)). The sliders should default with boundaries of \(-10\) and \(10\).

\[f(x)~=~a(x-h)^2+k\]

To make the animation shown below, two of the parameters were set to constants, and the other parameter was animated by pressing the "play" button. That animated parameter varies between \(-10\) and \(10\) by default.

Find the value of each parameter, or if the parameter varies then choose "animated".

• \(a~=~\) animated-11
• \(h~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910
• \(k~=~\) animated-10-9-8-7-6-5-4-3-2-1012345678910

• \(a=\) animated
• \(h=\) 2
• \(k=\) -1

Try it out in Desmos!

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A parabola is randomly generated by randomly choosing parameters \(a\), \(h\), and \(k\) in the vertex-form equation:

\[y~=~a(x-h)^2+k\]

Find the parameters.

• \(a=\) -3-2-1/2-1/3-1/4-1/91/91/41/31/223
• \(h=\) -6-5-4-3-2-10123456
• \(k=\) -6-5-4-3-2-10123456

How many \(x\)-intercepts does the parabola have? (Some might not show on the graph.)

• There are 0 x-intercepts.There is 1 x-intercept.There are 2 x-intercepts.

To find \(a\), you might use this reference. Notice, if \(a\) is negative, the parabola faces down, and if \(a\) is positive, the parabola faces up. If \(|a|\) is bigger than 1, then the parent \(y=x^2\) is stretched vertically. If \(|a|\) is less than 1, then the parent \(y=x^2\) is shrunk vertically. To find the exact value, you need to look at precisely where the curve intersects the lattice points.

To find \(h\) and \(k\), you can just find the coordinates of the vertex.

To determine the number of roots...

• If the vertex is on the \(x\) axis, there is one root.
• If the parabola opens up and the vertex is above the \(x\) axis, then 0 roots.
• If the parabola opens down and the vertex is above the \(x\) axis, then 2 roots.
• If the parabola opens up and the vertex is below the \(x\) axis, then 2 roots.
• If the parabola opens down and the vertex is below the \(x\) axis, then 0 roots.

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=\frac{-1}{4}(x-{2})(x-{6})\)\(y=\frac{-1}{4}(x-{2})(x+{6})\)\(y=\frac{-1}{4}(x+{2})(x-{6})\)\(y=\frac{-1}{4}(x+{2})(x+{6})\)\(y=\frac{-1}{9}(x-{2})(x-{6})\)\(y=\frac{-1}{9}(x-{2})(x+{6})\)\(y=\frac{-1}{9}(x+{2})(x-{6})\)\(y=\frac{-1}{9}(x+{2})(x+{6})\)\(y=\frac{1}{4}(x-{2})(x-{6})\)\(y=\frac{1}{4}(x-{2})(x+{6})\)\(y=\frac{1}{4}(x+{2})(x-{6})\)\(y=\frac{1}{4}(x+{2})(x+{6})\)\(y=\frac{1}{9}(x-{2})(x-{6})\)\(y=\frac{1}{9}(x-{2})(x+{6})\)\(y=\frac{1}{9}(x+{2})(x-{6})\)\(y=\frac{1}{9}(x+{2})(x+{6})\)

The correct answer is: \[y=\frac{-1}{4}(x-{2})(x-{6})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((2,0)\) and \((6,0)\). Thus, we could say \(r=2\) and \(s=6\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=\frac{-1}{4}(x-{2})(x-{6})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=\frac{-1}{3}(x-{5})(x-{1})\)\(y=\frac{-1}{3}(x-{5})(x+{1})\)\(y=\frac{-1}{3}(x+{5})(x-{1})\)\(y=\frac{-1}{3}(x+{5})(x+{1})\)\(y=\frac{-1}{9}(x-{5})(x-{1})\)\(y=\frac{-1}{9}(x-{5})(x+{1})\)\(y=\frac{-1}{9}(x+{5})(x-{1})\)\(y=\frac{-1}{9}(x+{5})(x+{1})\)\(y=\frac{1}{3}(x-{5})(x-{1})\)\(y=\frac{1}{3}(x-{5})(x+{1})\)\(y=\frac{1}{3}(x+{5})(x-{1})\)\(y=\frac{1}{3}(x+{5})(x+{1})\)\(y=\frac{1}{9}(x-{5})(x-{1})\)\(y=\frac{1}{9}(x-{5})(x+{1})\)\(y=\frac{1}{9}(x+{5})(x-{1})\)\(y=\frac{1}{9}(x+{5})(x+{1})\)

The correct answer is: \[y=\frac{-1}{9}(x+{5})(x-{1})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((-5,0)\) and \((1,0)\). Thus, we could say \(r=-5\) and \(s=1\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=\frac{-1}{9}(x+{5})(x-{1})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-2(x-{7})(x-{9})\)\(y=-2(x-{7})(x+{9})\)\(y=-2(x+{7})(x-{9})\)\(y=-2(x+{7})(x+{9})\)\(y=\frac{-1}{4}(x-{7})(x-{9})\)\(y=\frac{-1}{4}(x-{7})(x+{9})\)\(y=\frac{-1}{4}(x+{7})(x-{9})\)\(y=\frac{-1}{4}(x+{7})(x+{9})\)\(y=\frac{1}{4}(x-{7})(x-{9})\)\(y=\frac{1}{4}(x-{7})(x+{9})\)\(y=\frac{1}{4}(x+{7})(x-{9})\)\(y=\frac{1}{4}(x+{7})(x+{9})\)\(y=2(x-{7})(x-{9})\)\(y=2(x-{7})(x+{9})\)\(y=2(x+{7})(x-{9})\)\(y=2(x+{7})(x+{9})\)

The correct answer is: \[y=2(x+{7})(x+{9})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((-9,0)\) and \((-7,0)\). Thus, we could say \(r=-7\) and \(s=-9\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=2(x+{7})(x+{9})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-3(x-{7})(x-{1})\)\(y=-3(x-{7})(x+{1})\)\(y=-3(x+{7})(x-{1})\)\(y=-3(x+{7})(x+{1})\)\(y=\frac{-1}{9}(x-{7})(x-{1})\)\(y=\frac{-1}{9}(x-{7})(x+{1})\)\(y=\frac{-1}{9}(x+{7})(x-{1})\)\(y=\frac{-1}{9}(x+{7})(x+{1})\)\(y=\frac{1}{9}(x-{7})(x-{1})\)\(y=\frac{1}{9}(x-{7})(x+{1})\)\(y=\frac{1}{9}(x+{7})(x-{1})\)\(y=\frac{1}{9}(x+{7})(x+{1})\)\(y=3(x-{7})(x-{1})\)\(y=3(x-{7})(x+{1})\)\(y=3(x+{7})(x-{1})\)\(y=3(x+{7})(x+{1})\)

The correct answer is: \[y=\frac{1}{9}(x+{7})(x+{1})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((-7,0)\) and \((-1,0)\). Thus, we could say \(r=-7\) and \(s=-1\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=\frac{1}{9}(x+{7})(x+{1})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-2(x-{3})(x-{5})\)\(y=-2(x-{3})(x+{5})\)\(y=-2(x+{3})(x-{5})\)\(y=-2(x+{3})(x+{5})\)\(y=\frac{-1}{4}(x-{3})(x-{5})\)\(y=\frac{-1}{4}(x-{3})(x+{5})\)\(y=\frac{-1}{4}(x+{3})(x-{5})\)\(y=\frac{-1}{4}(x+{3})(x+{5})\)\(y=\frac{1}{4}(x-{3})(x-{5})\)\(y=\frac{1}{4}(x-{3})(x+{5})\)\(y=\frac{1}{4}(x+{3})(x-{5})\)\(y=\frac{1}{4}(x+{3})(x+{5})\)\(y=2(x-{3})(x-{5})\)\(y=2(x-{3})(x+{5})\)\(y=2(x+{3})(x-{5})\)\(y=2(x+{3})(x+{5})\)

The correct answer is: \[y=2(x+{3})(x+{5})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((-5,0)\) and \((-3,0)\). Thus, we could say \(r=-3\) and \(s=-5\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=2(x+{3})(x+{5})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-3(x-{3})(x-{5})\)\(y=-3(x-{3})(x+{5})\)\(y=-3(x+{3})(x-{5})\)\(y=-3(x+{3})(x+{5})\)\(y=\frac{-1}{2}(x-{3})(x-{5})\)\(y=\frac{-1}{2}(x-{3})(x+{5})\)\(y=\frac{-1}{2}(x+{3})(x-{5})\)\(y=\frac{-1}{2}(x+{3})(x+{5})\)\(y=\frac{1}{2}(x-{3})(x-{5})\)\(y=\frac{1}{2}(x-{3})(x+{5})\)\(y=\frac{1}{2}(x+{3})(x-{5})\)\(y=\frac{1}{2}(x+{3})(x+{5})\)\(y=3(x-{3})(x-{5})\)\(y=3(x-{3})(x+{5})\)\(y=3(x+{3})(x-{5})\)\(y=3(x+{3})(x+{5})\)

The correct answer is: \[y=3(x-{3})(x-{5})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((3,0)\) and \((5,0)\). Thus, we could say \(r=3\) and \(s=5\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=3(x-{3})(x-{5})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-3(x-{2})(x-{4})\)\(y=-3(x-{2})(x+{4})\)\(y=-3(x+{2})(x-{4})\)\(y=-3(x+{2})(x+{4})\)\(y=\frac{-1}{9}(x-{2})(x-{4})\)\(y=\frac{-1}{9}(x-{2})(x+{4})\)\(y=\frac{-1}{9}(x+{2})(x-{4})\)\(y=\frac{-1}{9}(x+{2})(x+{4})\)\(y=\frac{1}{9}(x-{2})(x-{4})\)\(y=\frac{1}{9}(x-{2})(x+{4})\)\(y=\frac{1}{9}(x+{2})(x-{4})\)\(y=\frac{1}{9}(x+{2})(x+{4})\)\(y=3(x-{2})(x-{4})\)\(y=3(x-{2})(x+{4})\)\(y=3(x+{2})(x-{4})\)\(y=3(x+{2})(x+{4})\)

The correct answer is: \[y=3(x-{2})(x-{4})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((2,0)\) and \((4,0)\). Thus, we could say \(r=2\) and \(s=4\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=3(x-{2})(x-{4})\]

A quadratic function, with two real roots, is graphed below.

What equation represents the function in factored form?

\(y=-2(x-{7})(x-{1})\)\(y=-2(x-{7})(x+{1})\)\(y=-2(x+{7})(x-{1})\)\(y=-2(x+{7})(x+{1})\)\(y=\frac{-1}{9}(x-{7})(x-{1})\)\(y=\frac{-1}{9}(x-{7})(x+{1})\)\(y=\frac{-1}{9}(x+{7})(x-{1})\)\(y=\frac{-1}{9}(x+{7})(x+{1})\)\(y=\frac{1}{9}(x-{7})(x-{1})\)\(y=\frac{1}{9}(x-{7})(x+{1})\)\(y=\frac{1}{9}(x+{7})(x-{1})\)\(y=\frac{1}{9}(x+{7})(x+{1})\)\(y=2(x-{7})(x-{1})\)\(y=2(x-{7})(x+{1})\)\(y=2(x+{7})(x-{1})\)\(y=2(x+{7})(x+{1})\)

The correct answer is: \[y=\frac{-1}{9}(x-{7})(x-{1})\]

You can determine \(a\) by using this reference sheet. You also need to remember that if \(a<0\), then the parabola opens downward.

The roots are at \((1,0)\) and \((7,0)\). Thus, we could say \(r=7\) and \(s=1\) (or the other way around). We put those parameters into factored-form equation, \(y=a(x-r)(x-s)\), to get...

\[y=\frac{-1}{9}(x-{7})(x-{1})\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=-0.2(x-{9})(x+{2})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -6.05-5.5-3.5-2.452.453.55.56.05
• \(w=\) -6.05-5.5-3.5-2.452.453.55.56.05
• \(k=\) -6.05-5.5-3.5-2.452.453.55.56.05

Identify the parameters.

\[a=-0.2\] \[r=9\] \[s=-2\]

Use the formulas.

\[h=\frac{(9)+(-2)}{2}=3.5\]

\[w=\frac{|(9)-(-2)|}{2}=5.5\]

\[k=-(-0.2)(5.5)^2=6.05\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=0.1(x+{2})(x+{1})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -1.5-0.5-0.225-0.0250.0250.2250.51.5
• \(w=\) -1.5-0.5-0.225-0.0250.0250.2250.51.5
• \(k=\) -1.5-0.5-0.225-0.0250.0250.2250.51.5

Identify the parameters.

\[a=0.1\] \[r=-2\] \[s=-1\]

Use the formulas.

\[h=\frac{(-2)+(-1)}{2}=-1.5\]

\[w=\frac{|(-2)-(-1)|}{2}=0.5\]

\[k=-(0.1)(0.5)^2=-0.025\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=-2(x-{1})(x+{4})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -12.5-4.5-2.5-1.51.52.54.512.5
• \(w=\) -12.5-4.5-2.5-1.51.52.54.512.5
• \(k=\) -12.5-4.5-2.5-1.51.52.54.512.5

Identify the parameters.

\[a=-2\] \[r=1\] \[s=-4\]

Use the formulas.

\[h=\frac{(1)+(-4)}{2}=-1.5\]

\[w=\frac{|(1)-(-4)|}{2}=2.5\]

\[k=-(-2)(2.5)^2=12.5\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=-0.1(x+{9})(x+{6})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -7.5-5.625-1.5-0.2250.2251.55.6257.5
• \(w=\) -7.5-5.625-1.5-0.2250.2251.55.6257.5
• \(k=\) -7.5-5.625-1.5-0.2250.2251.55.6257.5

Identify the parameters.

\[a=-0.1\] \[r=-9\] \[s=-6\]

Use the formulas.

\[h=\frac{(-9)+(-6)}{2}=-7.5\]

\[w=\frac{|(-9)-(-6)|}{2}=1.5\]

\[k=-(-0.1)(1.5)^2=0.225\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=3(x-{4})(x+{1})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -18.75-6.75-2.5-1.51.52.56.7518.75
• \(w=\) -18.75-6.75-2.5-1.51.52.56.7518.75
• \(k=\) -18.75-6.75-2.5-1.51.52.56.7518.75

Identify the parameters.

\[a=3\] \[r=4\] \[s=-1\]

Use the formulas.

\[h=\frac{(4)+(-1)}{2}=1.5\]

\[w=\frac{|(4)-(-1)|}{2}=2.5\]

\[k=-(3)(2.5)^2=-18.75\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=4(x+{8})(x-{4})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -144-16-6-22616144
• \(w=\) -144-16-6-22616144
• \(k=\) -144-16-6-22616144

Identify the parameters.

\[a=4\] \[r=-8\] \[s=4\]

Use the formulas.

\[h=\frac{(-8)+(4)}{2}=-2\]

\[w=\frac{|(-8)-(4)|}{2}=6\]

\[k=-(4)(6)^2=-144\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=0.5(x-{4})(x+{9})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -21.125-6.5-3.125-2.52.53.1256.521.125
• \(w=\) -21.125-6.5-3.125-2.52.53.1256.521.125
• \(k=\) -21.125-6.5-3.125-2.52.53.1256.521.125

Identify the parameters.

\[a=0.5\] \[r=4\] \[s=-9\]

Use the formulas.

\[h=\frac{(4)+(-9)}{2}=-2.5\]

\[w=\frac{|(4)-(-9)|}{2}=6.5\]

\[k=-(0.5)(6.5)^2=-21.125\]

Quadratic functions can be written in factored form:

\[y~=~a(x-r)(x-s)\]

where \(r\) and \(s\) represent the two \(x\)-intercepts and \(a\) represents the vertical stretch factor (and possibly a reflection if \(a\) is negative).

It can be helpful to use the following formulas, when graphing from factored form, to find the vertex:

\[h~=~\frac{r+s}{2}\] \[w~=~\frac{|r-s|}{2}\] \[k~=~-aw^2\]

The vertex is at \((h,k)\) and \(w\) represents the distance from either \(x\)-intercept to the axis of symmetry.

A quadratic function in factored form is shown below:

\[y=-0.5(x-{9})(x+{2})\]

Find \(h\), \(w\), and \(k\).

• \(h=\) -15.125-6.125-5.5-3.53.55.56.12515.125
• \(w=\) -15.125-6.125-5.5-3.53.55.56.12515.125
• \(k=\) -15.125-6.125-5.5-3.53.55.56.12515.125

Identify the parameters.

\[a=-0.5\] \[r=9\] \[s=-2\]

Use the formulas.

\[h=\frac{(9)+(-2)}{2}=3.5\]

\[w=\frac{|(9)-(-2)|}{2}=5.5\]

\[k=-(-0.5)(5.5)^2=15.125\]